Cusps and the Family Hyperbolic Metric

Cusps and the family hyperbolic metric

Scott A. Wolpert

March 13, 2007

Abstract The hyperbolic metric for the punctured unit disc in the Euclidean plane is singular at the origin. A renormalization of the metric at the origin is provided by the Euclidean metric. For Riemann surfaces there is a unique germ for the isometry class of a complete hyperbolic metric at a cusp. The renormalization of the metric for the punctured unit disc provides a renormalization for a hyperbolic metric at a cusp. For a holomorphic family of punctured Riemann surfaces the family of (co)tangent spaces along a puncture deﬁnes a tautological holomorphic line bundle over the base of the family. The Hermitian connection and Chern form for the renormalized metric are determined. Connections to the work of M. Mirzakhani, L. Takhtajan and P. Zograf, and intersection numbers for the moduli space of punctured Riemann surfaces studied by E. Witten are presented.

1 Comparing cusps

The renormalization of a hyperbolic metric at a cusp is introduced. The setting is used to present an intrinsic norm for the germ of a holomorphic map at a cusp. A compact Riemann surface R having punctures and negative Euler characteristic has a complete hyperbolic metric, [Ahl73]. The geometry of a cusp of a hyperbolic metric is standard. From the uniformization theorem for a puncture p there is a distinguished local conformal coordinate with z(p) = 0 and the metric locally given by the germ of

|dz| 2 ds2 = . (1) |z| log |z|

The distinguished coordinate is given as z = e2πiζ for the cusp represented at inﬁnity for the upper half plane H with coordinate ζ and translation

1 ζ → ζ + 1 . The canonical coordinate z is unique modulo a unimodular factor and the circle |z| = c is the closed horocycle about p with hyperbolic length ` = −2π/ log c. The complete hyperbolic metric for the punctured unit disc D = {0 < |z| < 1} is likewise given by formula (1). The unit area neighborhood of the puncture {`(z) ≤ 1}⊂D isometrically embeds to a neighborhood of each puncture of a complete hyperbolic metric, [Leu67]. A conformal coordinate ξ for a neighborhood of a puncture p can be used to describe the conformal completion at p. The omitted point p corresponds to the omitted value of ξ. The tangent space at p is described in terms of the tangent space at the omitted value of ξ. We use these general observations to define a norm for the tangent space at the puncture. Definition 1 Let R be a Riemann surface with hyperbolic metric and a puncture p with canonical local coordinate z as above. The canonical norm k ∂ k for the tangent space at p is defined by ∂z can =1. As noted the coordinate z is unique modulo a unimodular factor. For canonical coordinates z and w respectively for complete hyperbolic metrics 2 2 ds1 and ds2 consider a germ at the origin of a holomorphic map w = h(z)= zg(z) with h(0) = 0. We have for the ratio of metrics

0 2 0 2 ∗ 2 2 −1 |zh (z)| log |z| h (z) log |z| (h ds )(ds ) = = 2 1 |h(z)| log |h(z)| g(z) log |zg(z)|

log |h0(0)| 1 =1− 2 + O . (2) log |z| g (log |z|)2 We use the observations to present a formula for the norm of a map between tangent spaces. Consider R1 and R2 with complete hyperbolic metrics 2 2 ds1 and ds2 with punctures p1 and p2. For the germ h of a holomorphic map from a neighborhood of p1 to a neighborhood of p2 the canonical norms for the tangent spaces determine a norm for the differential dh. Observe that a point q close to p1 lies on a unique simple closed horocycle about the puncture. For q close to p1 write `(q) for the associated horocycle length. Lemma 2 Let h be the germ of a holomorphic map between punctures. At ∗ 2 2 −1 2 the puncture the differential satisfies lim (h ds2)(ds1) =1and log kdhkcan = q→p1 2π ∗ 2 2 −1 lim ` log((h ds2)(ds1) ). For the map h expressed in terms of canonical q→p1 0 coordinates w = h(z) the norm satisfies kdhkcan = |h (0)|. Proof. For canonical coordinates at the punctures the formulas follow from the observation ` = −2π/ log |z| and expansion (2). The proof is complete.

2 2π ∗ 2 2 −1 We will study the variation of lim ` log(h ds2)(ds1) for a family of Riemann surfaces. In the following sections we consider holomorphic families of Riemann surfaces. In preview, for ν(s) a suitable family of Beltrami differentials holomorphic in a parameter s, ν(s) with fixed compact support, a holomorphic family {Rν(s)} is defined. A conformal coordinate ζ for a neighborhood of a puncture p of R also serves as a conformal coordinate for Rν(s) provided the support of the coordinate is disjoint from supp(ν(s)). ∂ Accordingly the tangent vector ∂ζ is a holomorphic section of the family of tangent spaces along p. In this case the hyperbolic metrics are locally conformal and easily compared as ds2 = e2f ds2 . For the canonical coordinates Rν(s) R at p, z for R and w for Rν(s), the composition of w(ζ)◦(z(ζ)−1) is the germ of a holomorphic map h. We have from Lemma 2 that log kdhk2 = lim 4π f. can q→p ` We will study the limit in the following sections. We separately note that the Schwarz lemma also provides a comparison for germs of hyperbolic metrics at a puncture, [Ahl73]. For U a punctured 2 2 neighborhood of a puncture p with germs of hyperbolic metrics ds1 and ds2, 2 2 let V⊂Ube a neighborhood with ∂V compact in U. The ratio ρ = ds2/ds1 has limit unity at p from the comparison (2). In a local conformal coordinate ζ the constant curvature equation is −(ds2)−1∆ log ds2 = −1 for ds2 now the metric local expression and ∆ the coordinate Euclidean Laplacian. The 2 2 2 2 pair of metrics satisfy ∆ log ds2/ds1 = ds2 − ds1. The ratio ρ is continuous and has a maximum on V∪{¯ p}. Provided the maximum of ρ is greater than unity, the difference of metrics is positive at the maximum. In this case from the combined equations log ρ is strictly convex at the maximum and the maximum cannot be interior. It follows that the maximum of ρ is either 2 unity or occurs on ∂V. An application is provided for V the ds1 horoball neighborhoods {`(q) <`} of the cusp. The maximum of ρ on {`(q) ≤ `} is either unity or occurs only on {`(q)=`}. A general property follows. The maximum max{`(q)=`} ρ either has the constant value unity for all small `, or is strictly increasing in `. The analogous statement for min{`(q)=`} ρ is also valid.

2 The prescribed curvature solution

The solution of the prescribed curvature equation describes the ratio of a pair of hyperbolic metrics in a neighborhood of a puncture. For a family of hyperbolic metrics we consider the variation in parameters of the prescribed curvature solution. For calculation of the connection and Chern forms it is required that variations of the prescribed curvature solutions are suitably

3 bounded at the cusps. We find that adapted variations are suitably bounded. Our considerations begin with the basics of variations of conformal struc- tures and variations of solutions of the prescribed curvature equation. We provide the required formulas and estimates. Let R be a Riemann surface with hyperbolic metric ds2 and canonical line bundle κ. Let S(p, q), p, q ∈ Z be the space of smooth sections of κp/2⊗κq/2 and S(r)=S(r, −r). For a p-differential ϕ ∈ S(2p, 0) the product (ds2)−p/2ϕ is an element of S(p) and the conjugate (ds2)−p/2ϕ an element of S(−p). An element of S(r) has a well-defined absolute value. Metric- derivative operators (essentially the covariant derivatives) on S(r) are defined as follows: for ζ a local conformal coordinate and ds2 = ρ2(ζ)|dζ|2 then K = ρr−1 ∂ ρ−r and L = ρ−r−1 ∂ ρr, [Wol90]. Note for f ∈ S(r) that r ∂ζ r ∂ζ f ∈ S(−r), Kr : S(r) → S(r + 1), Lr : S(r) → S(r − 1) and Kr = L−r. Further note for f ∈ S(p), g ∈ S(r − p) that Kr(fg)=gKpf + fKr−pg and Lr(fg)=gLpf +fLr−pg. The hyperbolic metric Laplacian operator on functions is given as D =4L1K0.Forf ∈ S(r) the absolute value |f| is a function k and a C norm is defined by kfk0 = sup |f| and kfkk = P|P |≤k kPfk0 for the sum over all products of operators K∗ and L∗ of length at most k. We are also interested in functions suitably small in the cusps. A weighted-normed −n space C`n is defined for f ∈ S(r)bykfk`n = sup`≤1 |` f|∨supR−{`≤1} |f|. In effect from Lemma 2 we are interested in the variational formulas for a family of hyperbolic metrics with variational terms bounded in C`. Let R be a punctured Riemann surface and Q(R) the associated space of holomorphic quadratic differentials with at most simple poles at punctures. The space B(R) of continuous Beltrami differentials is the finite C0-norm elements of S(−2). A basic feature of Riemann surfaces is the integral ∈ ∈ pairing RR µϕ for µ B(r) and ϕ Q(R). The deformation space of R is the Teichmüllerspace T , [Ahl61, Ear77, Nag88]. We will write Tg,n for the Teichmüllerspace of genus g, n punctured surfaces. At R the holomorphic tangent space to the deformation space is B(R)/B(R)⊥ and the holomorphic cotangent space is Q(R). A new conformal structure Rµ for µ ∈ B(R) with kµk < 1 is defined as follows. For z a local conformal coordinate for R and w(z) a local homeo- morphism solution on the range of z for the Beltrami equation wz¯ = µwz the µ composition w◦z is a local conformal coordinate for R . In brief for {zα}α∈A an atlas for the R-conformal structure, the compositions {wα ◦ zα}α∈A provide an atlas for the Rµ-conformal structure. Local holomorphic coordinates for T are described as follows: for {ν1,...,νm} continuous Beltrami differ- ⊥ m entials spanning B(R)/B(R) and s ∈ C , set ν(s)=Pj sjνj ; for s small

4 s → Rν(s) is a local holomorphic coordinate for T , [Ahl61, Ear77, Nag88]. The harmonic Beltrami differentials H(R)={µ ∈ B(R) | K−2µ =0} form a natural subspace. The interest with harmonic Beltrami differentials arises from the observation that variational formulas are simplified by their introduction, [Ahl61, Nag88, Wol89, Wol86]. Basic properties are as follows: H(R) ⊂ B(R) is a direct summand for B(R)⊥ and consequently H(R) ' B(R)/B(R)⊥; the mapping ϕ → (ds2)−1ϕ¯ is a complex anti linear bijection of Q(R)toH(R). The considerations of the first section require maps holomorphic in a neighborhood of the punctures. To this purpose we introduce smooth trun- cations of elements of H(R). Begin with a choice of local coordinate z for a neighborhood U of the puncture p with z(p) = 0. Choose a function χ on C to be an approximate characteristic function of {|z| > 1} such that χ is smooth, vanishes identically on {|z|≤1/2} and is identically unity on {|z|≥1}.For sufficiently small the -truncation of µ ∈H(R) at the puncture p is locally defined as χ(z/)µ(z); as tends to zero the truncation tends to µ. The truncation of µ is defined by introducing a truncation at each puncture. For µ harmonic K−2χµ = µK0χ and we combine local expressions at a puncture for the hyperbolic metric and a holomorphic 4 quadratic differential to find that kK−2χµk1 is O( log ) for tending to zero. To compare metrics, we introduce a particular pullback of a metric. For 2 a Riemann surface S with almost complex structure JS, JS = −id; arbitrary 2 2 metric dσ and Kählerform ω we note that dσ =2ω( ,JS ). For a Riemann 2 surface R with almost complex structure JR, JR = −id, and h : R → S a 2 ∗ smooth map the J-pullback by h metric is defined as dσ∗ =2h ω( ,JR ). The J-pullback metric has a straightforward local coordinate description. For z a conformal coordinate for R, w a conformal coordinate for S and 2 2 2 2 2 2 2 dσ =(α(w)|dw|) then dσ∗ = α(w(z)) (|wz| −|wz¯| )|dz| . As explained in [Wol90, pg. 449], introduction of the J-pullback metric simplifies various variational formulas and in particular gives rise to formulas which are local in the Beltrami differential and its derivatives. By comparison introduction of ∗ 2 the standard pullback h dσ entails solving the potential equation Fz¯ = µ. 2 To compare hyperbolic metrics we begin with dsR the hyperbolic metric 2 of R and a smooth map h : R → S. Introduce now dσ∗ the J-pullback by −1 2 h metric of dsR to S. (Use of the inverse map will simplify the resulting 2 expressions.) The metric dσ∗ on S is a perturbation of the hyperbolic metric 2 on R. Let C be the Gaussian curvature of dσ∗ and C∗ = C ◦ h. Further let 2 Dσ∗ be the dσ∗-Laplacian on S and D∗ its pullback by h, i.e. for g a smooth function on S then (Dσ∗g) ◦ h = D∗(g ◦ h).

5 We review the formulas for C∗ and D∗ needed for the present considerations. The complete variational formulas were presented in [Wol90, Sec. 5.2, esp. Lemma 5.3]. We will consider a family {S = Rµ(a)} for µ(a)=aµ for a Beltrami diﬀerential µ ∈ B(R) and a real parameter a. The curvature C∗(a) and pullback-Laplacian D∗(a) depend real analytically on a, [Wol90]. Basic properties are as follows. In general for supp(µ) ⊂C, we have on the complement R −C that C∗(a)=−1 and (for local expressions) D∗(a)=D. The variational derivatives of C∗ and D∗ for µ(a)=aµ at a = 0 are as follows 2 Ca =4

Da = −8

The formulas simplify for harmonic Beltrami diﬀerentials since K−2µ vanishes and are similarly approximated for truncated diﬀerentials since K−2µ is O( log4 ). 2 We are ready for h : R → S to compare metrics, beginning with dsR the complete hyperbolic metric of R. The complete hyperbolic metric of S is −1 2 described in terms of the J-pullback by h metric dσ∗ and the solution of the prescribed curvature equation 2 2f˜ 2 ˜ 2f˜ dsS = e dσ∗ for Dσ∗f − C = e , [Wol90, pg. 453].

As noted above the composition f = f˜◦ h satisfies the equation D∗f − C∗ = 2f ˜ 2f˜ 2f˜ 2 e . The equation Dσ∗ f − C = e is equivalent to the curvature of e dσ∗ being identically −1, [Wol90]. The uniformization theorem and estimates of the first section provide that the prescribed curvature equation has a unique solution in C`, [Ahl73]. Standard results from Teichmüllertheory provide that for µ(a) ∈ B(R) varying real analytically the prescribed curvature solution f(a) varies real analytically in Ck-norm for compact subsets of R, [AB60, Ear77]. We are now ready to consider the variation of the prescribed curvature solution. The expressions for the Laplacian D∗(a), curvature C∗(a) and solution f(a) are real analytic in the parameter a. The expressions for the variations in a are determined. We have for small values of a the equations 2f 2f D∗f − e = C∗,D∗fa + Daf − 2e fa = Ca (5) and after adding Dfa to each side of the second equation 2f Dfa − 2e fa =(D − D∗)fa − Daf + Ca. (6)

6 For a Beltrami differential µ with compact support the right hand side of the third equation has compact support. We now use the equations to show that f(a) is a differentiable C`-valued function. For ` the horocycle length −n function, the C`n weighted-norm is kfk`n = sup`≤1 |` f|∨supR−{`≤1} |f|. Lemma 3 Let f(a) be the prescribed curvature solution for hyperbolic metrics and Beltrami differential µ(a)=aµ, µ with compact support. The solution f(a) is a differentiable C`-valued function of the parameter a.For small values of a the solution satisfies 2 2 f(a)=afa(0) + O(a ) and fa(a)=fa(0) + afaa(0) + O(a ) for remainder terms bounded in C`. Proof. We iterate basic bounds to establish the desired expansions. In preview, a bound for f gives a bound for e2f which from equation (6) gives a bound for the derivative fa and from integration in the parameter an improved bound for f. We start by noting from the final paragraph of the first section that kfk0 is bounded by supsupp(µ) |f| and as noted above on supp(µ) the solution f is real analytic in Ck in a. Lemmas 11 and 12 below will be used to pass from estimates on supp(µ) for the right hand side of (6) to estimates on R. To begin, from Lemma 11 and the bounds for quantities on the compact support of the right hand side of (6) it follows from Lemma 12 that fa is uniformly bounded in C` for all suitably small a. We next 00 a − 0 integrate in the parameter Ra0 fa da to find for all small a, a a that f(a) 0 0 is O(a)inC` and that f(a)=f(a )+O(|a − a |)inC`. The expansion is the Taylor expansion for a continuous C`-valued function. We next compare the equations (6) for values a and a0 with e2f(a) = 0 0 e2f(a )+O(|a−a |) and apply Lemmas 11 and 12 to find for all small a, a − a0 0 0 that fa(a)=fa(a )+O(|a − a |)inC`. We next integrate in the parameter 0 0 0 0 2 to find that f(a)=f(a )+(a−a )fa(a )+O((a−a ) )inC`. The expansion is the Taylor expansion for a differentiable C`-valued function. We next substitute the first-order expansions for the right hand side 2f(a) 2 rhs(a) of (6) and also the expansion e =1+2afa(0) + O(a )inC` to 2 find the equation in C` (modulo terms of order a in C`2 ) for fa(a) Df − 2f − 2af f = rhs(0) + arhs (0) + O (a2). a a a a a C`2 The resulting equation f =(D − 2)−1 rhs(0) + arhs (0) + 2af (0)2 + O (a2) a a a C`2 provides for the desired Taylor expansion for fa(a). We note in passing that all remainder terms depend on supp(µ). The proof is complete.

7 3 The canonical curvature

We present the formula for the Hermitian connection and curvature of the canonical norm for the family of tangent spaces along a puncture. The curvature is given in terms of the Takhtajan-Zograf form. Let Γ be a Fuchsian group uniformizing the Riemann surface R with n punctures. Let Γ1,...,Γn be a set of non-conjugate maximal parabolic subgroups of Γ representing the punctures. Further let σj ∈ PSL(2; R)be −1 transformations such that σj Γj σj =Γ∞ with Γ∞ the group of integer- translations. The Eisenstein series for the jth puncture of R is deﬁned for ζ ∈ H and ~~1as~~

~~−1 s Ej(ζ,s)= X =(σj γζ) ,j=1,...,n, [Bor97, Ven90]. γ∈Γj\Γ~~

The special value Ej(ζ)=Ej(ζ,2) plays an important role for deformation theory, [OW07, TZ91, Wol94]. The Fourier expansions of the Eisenstein series have a simple form, th [Bor97, Ven90]. Conjugate the k cusp to inﬁnity and consider Ej(σk(ζ)) = Ej(σk(ζ), 2) which is Γ∞ invariant. For ζ = x + iy, the Kronecker delta δ, and coeﬃcients cjk, there is the expansion for large y 2 −1 −2πy Ej(σk(ζ)) = δjky + cjky + O(e ).

The functions Ej(ζ) satisfy the differential equation DE =2E and are not elements of L2(R). The hyperbolic metric Laplacian D acting on L2(R) has non positive spectrum. The Green’s function G(ζ,ζ0) for the operator (D − 2)−1 is a part of our considerations. The Green’s function is the integral kernel with respect to the hyperbolic area element dA for inverting the (D−2) operator: 0 0 − ∈ 0 for u(ζ)=RR G(ζ,ζ )g(ζ )dA then (D 2)u = g for g C . We again conjugate the jth cusp to infinity and consider Fourier expansions. For ζ0 fixed, ζ = x + iy, there is the expansion for y large 0 0 −1 −2πy G(σj(ζ),ζ )=cj(ζ )y + O(e ) and for g with compact support −1 −2πy u(σj(ζ)) = cj(u)y + O(e ). We begin considerations with the approach of L. Takhtajan and P. Zograf to express the leading coefficient of (D − 2)−1g in terms of the integral

~~RR gE dA, [TZ91, Lemma 2]. The approach follows the argument for the Maass-Selberg relation, [Bor97, Ven90].~~

8 Lemma 4 Let Γ be a Fuchsian group uniformizing a Riemann surface with punctures. For g with compact support and `j the horocycle length function for the jth puncture, the operator (D − 2)−1 satisﬁes

−1 −1 −1 Z lim `j ((D − 2) g)(σjζ)= gEj dA. =ζ→∞ 3 R Proof. We begin with the deﬁning propertyg ˆ =(D − 2)ˆu and observe that Z Z Z Ej ◦σjgdA= Ej ◦σj Du−2Ej ◦σjudA= Ej ◦σjDu−DEj ◦σjudA R R R

(where we write g forg ˆ◦σj and u foru ˆ◦σj .) The terms of the third integrand are not individually integrable since the leading coefficient of Ej ◦ σju at infinity is =ζ. To consider the third integral we introduce a fundamental −1 domain F for σj Γσj, containing the cusp neighborhood {0 ≤<ζ<1, =ζ> 1}. We further introduce the sub domains F Y = {ζ ∈F|=ζ ≤ Y } and from the above Z Z Ej ◦ σjgdA= lim Ej ◦ σjDu − DEj ◦ σjudA R =Y →∞ FY and apply Green’s formula to find

Z ∂u ∂Ej ◦ σj = lim Ej ◦ σj − uds =Y →∞ ∂FY ∂n ∂n ∂ F Y ∩ F for the hyperbolic metric elements ∂n and ds. The integral over ∂ ∂ vanishes by consideration of the orientation and the group invariance of Ej ◦ σj and u. The Fourier expansions for Ej ◦ σj and u provide that the −Y remaining integral over {0 ≤<ζ<1, =ζ = Y } is −3 cj(û)+O(e ). The proof is complete. The tangent space to the deformation space at a Riemann surface R is represented by the space H(R) of harmonic Beltrami differentials. A Hermitian form for H(R) defines a Hermitian metric for Teichmüllerspace. For µ, ν ∈H(R) and hyperbolic area element dA the Weil-Petersson (WP) form is Z hµ, νiWP = µν¯ dA, [Ahl61] R and for R with punctures p1,...,pn the Takhtajan-Zograf (TZ) form for the puncture pj is Z hµ, νiT Z,pj = µνE¯ j dA, [TZ91]. R h i The Takhtajan-Zograf metric is Pj , T Z,pj . We are ready to present the formula for the Hermitian connection and curvature for the canonical norm.

9 Theorem 5 Let T be the Teichmüller space of a Riemann surface R with punctures. The canonical norm kkcan,p for the family of tangent spaces along the puncture p has Hermitian connection vanishing on H(R) and Chern form 2i c1(kkcan,p)= 3 h , iT Z,p on H(R). Proof. We combine considerations. We have from the discussion following Lemma 2 that for the family {Rν(s)}, ν(s) a truncated Beltrami differential with compact support, and ζ a local conformal coordinate at a puncture, that log k ∂ k2 = lim 4π f for ds2 = e2f ds2 . The hyperbolic metrics ∂ζ can q→p ` Rν(s) R of {Rν(s)} are described in terms of the solutions f(s) of the prescribed curvature equation. The Taylor expansions of f(s) of Lemma 3 provide that 4π 2 4π limq→p ` f(s)isC at s = 0 with initial s-derivative limq→p ` fs(0) and 4π initial ss¯-derivative limq→p ` fss¯(0). From equation (5) and Lemma 4 we find in terms of the curvature, Laplacian and prescribed curvature solution the variations at s =0

~~4π 4π −1 −4π Z lim fs(0) = lim (D − 2) Cs = CsEp dA q→p ` q→p ` 3 R and~~

~~4π 4π −1 lim fss¯(0) = lim (D − 2) (Css¯ − Dsfs¯ − Ds¯fs +4fsfs¯) q→p ` q→p ` −4π Z = (Css¯ − Dsfs¯ − Ds¯fs +4fsfs¯)Ep dA 3 R~~

(the formulas for fs(0) and fss¯(0) are a straightforward calculation.) We next observe that since the canonical norm is twice differentiable its Hermi- tian connection 1-form Θ and curvature 2-form Ω have well-defined evaluations on H(R). The evaluations are given by the limit of the truncated Bel- trami differentials introduced in the prior section (the limit as the parameter tends to zero.) We have from the estimate for truncated Beltrami differentials combined with (3) and (4) for elements of H(R) the resulting formulas − −1 −1 | |2 at s =0:Cs =0,fs =(D 2) Cs = 0 and Css¯ = 2 D µ . As the final step −1 | |2 − | |2 we apply Green’s formula to note that 2 RR D µ Ep dA = RR µ Ep dA and finally that: 4π i 2i Θ(µ)=0, Ω(¯µ,µ)= hµ, µi and c (kk )= Ω= h , i . 3 T Z,p 1 can,p 2π 3 T Z,p The proof is complete. h i i h i Associated to a Hermitian form , is a (pre) Kählerform 2 , . The above result provides a new proof that the TZ metric is Kähler,[TZ91]. We i h i now write ωT Z,p = 2 , and restate the result.

10 Corollary 6 The Chern form of the canonical norm for the family of tan- kk −4 gent spaces along a puncture p satisﬁes c1( can,p)= 3 ωT Z,p. Takhtajan-Zograf were able to determine the Chern form for a single puncture without identifying the metric [TZ91, formula (11)], while L. Weng using Arakelov theory determined the Chern form for multiple punctures without identifying the metric [Wen01].

~~4 Applications~~

We present beginning properties of the canonical norm Chern form and connections to the work of other authors. The TZ metric is Kähler,[TZ91] and incomplete, [Obi99]. K. Obitsu, W. K. To and L. Weng [OTW06] have recently determined the asymptotic behavior of the metric akin to the original result of H. Masur, [Mas76]. A simple property of the metric comes = 2 from the observation E = Pγ∈Γ∞\Γ (γz) and that the integral RR µνE¯ dA can be unfolded to µν¯(=ζ)2 dA which is a special value of the Rankin- RΓ∞\H Selberg convolution L-function, [TZ91, Theorem 2]. The families of cotangent spaces along punctures give rise to rational cohomology classes on the moduli space Mg,n of genus g, n punctured Rie- mann surfaces. By definition ψj is the rational Chern class of the orbifold line bundle whose fiber at the point [R; p1,...,pn] ∈Mg,n is the cotangent space at pj (see [AC96] for the definition on Mg,n.) The canonical norm provides a metric for the line bundles ψj; for the canonical local coordinate z at a puncture, kdzkcan = 1. We restate our main formula in the present setting.

Corollary 7 The Chern form of the canonical norm for the family of cotan- kk 4 gent spaces ψp along a puncture p satisfies c1( can,p)= 3ωT Z,p. L. Takhtajan and P. Zograf used Quillen’s metric to calculate the first Chern form of the determinant line bundle for families of ∂¯-operators, [TZ91]. The authors considered the Teichmüllerspace Tg,n, the Teichmüllercurve Cg,n → −k th Tg,n and Ek = Tvert Cg,n the k symmetric power of the dual of the vertical line bundle of Cg,n →Tg,n. On a fiber Cg,n →Tg,n of the Teichmüller curve the vertical tangent space Tvert Cg,n coincides with the tangent space ¯ of the fiber. Associated to the family of ∂-operators for Ek is an index ¯ ¯ bundle ind ∂k and determinant holomorphic line bundle det ind ∂k with a Quillen metric kkQuillen determined from the hyperbolic metric of Riemann surfaces. Takhtajan-Zograf found a local index formula using [Wol86] for

~~11 families of compact Riemann surfaces with 2g − 2 > 0, k ≥ 0,~~

~~6k2 − 6k +1 c (det ind ∂¯ )= ω , [TZ87] 1 k 12π2 WP and for families of punctured Riemann surfaces with 2g − 2+n>0, k ≥ 0,~~

~~6k2 − 6k +1 1 c (det ind ∂¯ )= ω − X ω , [TZ91]. 1 k 12π2 WP 9 T Z,pj j~~

Quillen’s metric involves the zeta function determinant of the Laplacian. For punctured Riemann surfaces Takhtajan-Zograf used special values of the Selberg zeta function in place of zeta function determinants, [TZ91, formula (6)]. In [Wol90] the family hyperbolic metric for the vertical line bundle Tvert Cg →Tg was used to find a Chern form on Cg and to calculate the pushdown of the square of the form. We found that the pushdown class κ1 1 is represented by the pushdown form π2 ωWP (see the section below on the WP Kählerform.) By using truncated harmonic Beltrami differentials the formula can be generalized to families of punctured Riemann surfaces. We now combine results and present a local form of the above Takhtajan-Zograf formula.

Corollary 8 For bundles over Tg,n the Quillen metric, vertical line bundle metric and cotangent spaces along punctures metric determined from the hyperbolic metric there is a pointwise relation of Chern forms

~~¯ 2 12 c1(det ind ∂k)=(6k − 6k +1)c1(κ1) − X c1(ψpj ). j~~

Certain comments are in order. First, the hyperbolic metric is determined by a choice of conformal structure and does not involve a choice of marking and so the above considerations are valid for Mg,n the moduli space of punctured Riemann surfaces. Second, the considerations for the compactiﬁed moduli space of stable curves Mg,n have not been eﬀected. An application is the curvature of the conormal bundle to the divisor of noded Riemann surfaces. Families of cotangent spaces along punctures can be used to describe the conormal bundle. In particular a pair of families of punctured Riemann surfaces {R} and {R0} and a formal pairing of the punctures p of R and p0 of R0 determines a family D = {R ∨ R0} of noded Riemann surfaces, [Ber74]. Consider the family M where the node p ∨ p0 is allowed to open. The product of cotangent spaces along p and p0

12 defines a line bundle λ over D⊂M. We recall that λ is isomorphic to the conormal bundle of the divisor D⊂M. The family M of noded Riemann surfaces can be described in terms of deformations supported away from the node and the plumbing family {(z,w,t) | zw = t, |z|, |w|, |t| < 1}→{|t| < 1}. For such a description the function t becomes a local defining function for the divisor D⊂M. For a change of parameterization f(z),g(w) and h(t) for the plumbing family with f(0),g(0) and h(0) each zero there is the basic relation f 0(0)g0(0) = h0(0). The relation provides the cocycle relation for the isomorphism of the conormal bundle λ and the product of cotangent spaces along punctures. The product of canonical norms for cotangent spaces provides a norm for λ, as well as for the inverse bundle −1 −1 λ . From Corollary 6 the curvature c1(λ ) is negative definite and we find a local form of the principle that the opening of a node is negative. L. Weng studied for punctured Riemann surfaces the intersection product for metrized line bundles and also the Deligne-Riemann-Roch isometry, [Wen01]. He introduced the metrized WP, TZ and logarithmic Mum- ford line bundles over Mg,n and determined first Chern forms. As part of his results he showed for the metrized TZ line bundle ∆TZ on Mg,n that 4 c1(∆TZ)= 3 ωTZ, [Wen01, pg. 278]. An application is for the volume of moduli spaces. M. Mirzakhani has considered the moduli space Mg,n(b1,...,bn) of genus g bordered Rie- mann surfaces with geodesic boundary components of prescribed length 1 ∧ (b1,...,bn), [Mir07a]. The Kählerform ωWP = 2 P d` dτ provides a symplectic form on Mg,n(b1,...,bn). Mirzakhani developed a recursive scheme for determining the volumes. Using an identity for geodesic length she es- tablished a general volume-result, [Mir07a].

Theorem 9 The volume Vg,n(b)=Vol(Mg,n(b1,...,bn)) is a polynomial 2 2 in the squares of geodesic boundary lengths b1,...,bn with 2α Vg,n(b)= X cg(α)b |α|≤3g−3+n n where α ranges over multi indices of (Z≥0) and cg(α) are positive values of π6g−6+2n−2|α|Q. Mirzakhani applied the result in [Mir07a] to provide a volume-expansion for tubular neighborhoods of the compactiﬁcation divisor D in the Deligne- Mumford compactiﬁed moduli space M. n The orbifolds Mg,n(b1,...,bn) form an (R≥0) bundle over Mg,n the moduli space of genus g, n punctured Riemann surfaces. A Riemann surface with geodesic boundaries and a point on each boundary is alternately

13 described by an n punctured Riemann surface and a product of n factors of S1, a principal torus-bundle over a punctured Riemann surface. Mirzakhani ﬁnds in [Mir07b] that symplectic reduction (for an (S1)n quasi-free action following Guillemin-Sternberg [Gui94]) can be used to provide a simple description for the family of K¨ahlerforms

~~2 bj ωWP = ωWP + X c1(ψj) (7) Mg,n(b1,...,bn) Mg,n 4 j~~

(the relation is for cohomology classes on M.) (Mirzakhani considers the symplectic form 2 ωWP and so formulas differ by a factor of 2.) The expansion is presented for a general choice of principal connection for the S1 bundles. An explicit principal connection is given by introducing the line bundles ψj, the canonical norms kkcan,j and Hermitian connections. The consequence is a local form of the above expansion. The local expansion agrees with a perturbation formula of K. Obitsu and the author, [OW07]. We considered the perturbation of the WP metric and Kählerform for the tangent subspaces parallel to the compactification divisor D⊂M. The expansion is a refinement to the work of H. Masur [Mas76], G. Daskalopou- los and R. Wentworth [DW03], and the author [Wol03]. For a family {R`} of hyperbolic surfaces given by pinching short geodesics all with common length `, we found for the Kählerforms restricted to the tangent subspaces parallel to the compactification divisor

~~`2 ωtgt (`)=ωtgt (0) + X ω (0) + O(`3) WP WP 3 T Z,pj j and with Corollary 7 the pointwise relation~~

~~`2 = ωtgt (0) + X c (ψ )+O(`3). WP 4 1 pj j~~

~~Mirzakhani combined her integration scheme and formula (7) to show that the collection of integrals (intersection pairings)~~

Z α1 αn 3g−3+n−|α| c1(ψ1) ···c1(ψn) ωWP Mg,n satisﬁes the recursion for the string equation and the dilaton equation. The intersection numbers combine to provide a partition function F for two- dimensional quantum gravity, [Wit91, Wit92]. E. Witten conjectured that

14 F F e would satisfy the KdV equations Lk e =0,k ≥−1 with Virasoro con- straint relations [Lm, Lk]=(m − k)Lm+k. M. E. Kazarian and S. K. Lando [KL06], Y.-S. Kim and K. Liu [KL05], M. Kontsevich [Kon92], M. Mulase and B. Safnuk [MS06], A. Okounkov and R. Pandharipande [OP01], and Mirzakhani [Mir07b] have veriﬁed the conjecture. The authors show that the Virasoro relations determine the intersection numbers of tautological line bundles. Mirzakhani’s integration scheme also determines the intersection numbers. A consequence of Corollary 7 is the following.

~~Corollary 10 The Virasoro relations or Mirzakhani’s integration scheme can be used to determine all TZ-WP pairings~~

~~Z α1 αn 3g−3+n−|α| ωTZ,1 ···ωT Z,n ωWP . Mg,n~~

~~We illustrate the result with an example. Mirzakhani provided in [Mir07a] the expansion for 2 ωWP for (g, n)=(0, 4) 1 V (b)= (4π2 + b2 + ···+ b2) 0,4 2 1 4 which corresponds to the integrals~~

~~2 Z 2 Z π = ωWP = π κ1 M0,4 M0,4~~

~~1 using that κ1 = π2 ωWP and from (7) to the integrals~~

Z c1(kkcan,j )=1. M0,4 The values agree with the evaluations of E. Arbarello and M. Cornalba [AC96] and P. Zograf [Zog93]. We apply the last evaluation to ﬁnd the TZ volume of M0,4. The volume form is dVTZ = Pj ωT Z,j and consequently

~~Z dVTZ =3. M0,4~~

~~5 The WP K¨ahlerform~~

~~The characteristic class of the WP K¨ahlerform is part of the present considerations. We now revisit our earlier treatment of the WP symplectic and~~

15 Kählerforms to find that ωW P,symplectic =2ωWP,Ka¨hler. Certain earlier formulas especially for integrals and characteristic classes need to be adjusted for the present considerations. An underlying real tangent space V for a complex manifold M has an almost complex structure J, J2 = −id. The complexification V C of the tangent space V is decomposed into the ±i eigenspaces of J. The decomposition is given as V 1,0 ⊕ V 0,1 with V ⊂ V C the subspace fixed by complex conjuga- tion. For a Riemann surface R at the corresponding point of the Teichmüller space the holomorphic tangent space is V 1,0 'H(R) and the holomorphic cotangent space is (V 1,0)∗ ' Q(R). The tangent-cotangent pairing for µ H in (R) and ϕ in Q(R)is(µ, ϕ)=RR µϕ. On the holomorphic tangent 1,0 space V 'H(R) the WP Hermitian form is h , iWP and the Kählerform i h i i ∧ ··· ∧ ωWP = 2 , WP (corresponding to the Kählerform 2 (dz1 dz¯1 dzm dz¯m) m on C .) We will see below that in effect ih , iWP was used in our earlier papers [Wol82, Wol83c, Wol83b, Wol83a, Wol86, Wol90]. Takhtajan-Zograf i h i use the Kählerform 2 , WP, [TZ91, pg. 402]. We studied in [Wol82] for a closed geodesic α on a Riemann surface the relationship between the Fenchel-Nielsen infinitesimal twist deformation tα and the geodesic-length function `α.Forθα the classical Petersson theta series for α, the infinitesimal Fenchel-Nielsen twist tα is represented by the i −1 harmonic Beltrami differential π (dA) θα in H(R)(dA the hyperbolic area element), [Wol82, Corollary 2.8]. F. Gardiner’s formula for the differential 2 < i h i of geodesic-length is d`α = π RR µθα, [Gar75]. A Kählerform 2 , is evaluated with a sum over permutation of vectors. We calculate the twist- length duality

~~2 Z i −1 (µ, d`α)= < µθα =2~~

The twist-length duality in terms of real tangent vectors is d`α =2ωWP,Kahler¨ ( ,tα). If we write ωW P,symplectic for the symplectic form used in our earlier papers then we have ωW P,symplectic =2ωWP,Kahler¨ ( , ), [Wol82, see The- orem 2.10]. The deﬁnition of the symplectic form and twist-length duality formula continued in our subsequent papers. There are corresponding adjustments to subsequent formulas. Integration formulas are relevant for the present considerations. The M1,1 and M0,4 area formulas become 2 2 ω = π and 2 ω =2π2 since the twist-length RM1,1 WP,Kahler¨ 6 RM0,4 WP,Ka¨hler duality formula was used to derive the integrand. To represent the appropri- 1 M ate characteristic class there is a further factor of 2 for 1,1 since the WP pairing should be for integration over tori modulo their elliptic involution. 1 ∧ The canonical coordinates formula becomes ωWP,Kahler¨ = 2 Pj d`j dτj

16 [Wol85, Theorem 1.3]. The characteristic class formula [Wol83a, formula (5.1)] should also be adjusted for the definition of the Kählerform. The formula for κ1 the pushdown of the square of the Chern form for the family hyperbolic metric is likewise affected [Wol86, see proof of Corollary 5.11] 1 and [Wol90]. The updated formula κ1 = π2 ωWP,Kahler¨ for [Wol86] and updated formula (5.1) for [Wol83a] now combine to agree with the formula κ1 =12λ − δ of D. Mumford, [Mum77] and the Takhtajan-Zograf κ1 calculation [TZ91, pg. 424].

~~6 Estimates for Green’s operators~~

~~We provide estimates for the operator (D − k)−1 acting on functions small at the cusps.~~

~~0 −1 Lemma 11 For k ∈ C with a positive inﬁmum mk the operator (−D+k) is continuous on L2(R). For a continuous g ∈ L2(R) the operator satisﬁes −1 −1 |(−D + k) g|≤(−D + mk) |g|.~~

Proof. For β the supremum of k we write k = β − kˆ and introduce the factorization (−D +k)−1 =(1−(−D +β)−1kˆ)−1(−D +β)−1. By hypothesis ˆ mk = β − sup k is positive and we can consider the geometric series

~~(1 − (−D + β)−1kˆ)−1 =1+(−D + β)−1kˆ +((−D + β)−1kˆ)2 + ··· (8)~~

(kˆ is now the multiplication operator.) From the spectral theorem the L2- norm of the operator (−D+β)−1 is β−1 and the above series of L2-operators converges since β−1 sup k<ˆ 1. The operator (−D + k)−1 is deﬁned on L2. Next we consider the pointwise behavior of (−D + c)−1g, for a continuous g ∈ L2. The integral kernel for (−D + c)−1, c a positive constant, is positive and so for a non negative continuous function v ∈ L2 we have the inequality −1ˆ −1 −1 (−D +β) kv≤ (−D +β) (β −mk) v. To estimate (−D +k) g we apply the right hand side of (8) to (−D + β)−1|g|≥|(−D + β)−1g| and apply the above inequality for each factor of (−D + β)−1kˆ. The result is a convergent −1 geometric series expansion for (−D + mk) |g|, the desired upper bound. The proof is complete. −1 We now consider the operator (D − 2) acting on functions in C`n ,n> 1. The Green’s function for the operator (−D + 2) is given as an absolutely convergent sum G(z, z0)=X Q(z, γz0) γ∈Γ

17 for δ(z, z0) hyperbolic distance and Q(z, z0)=−Q2(δ(z, z0)) for Q2 an associated Legendre function, [Fay77, Chap. 1]. We consider that Γ contains the group of integer translations Γ∞ as a maximal parabolic subgroup and introduce

~~G∞(z, z0)= X Q(z, γz0) and G†(z, z0)=G(z, z0) − G∞(z, z0). γ∈Γ∞~~

We write C = {` ≤ 1}⊂R for the union of the unit area horocycle regions and C∞ for the unit area horocycle region at inﬁnity. In the following we analyze the behavior of RR G(z, z0)g(z0) dA by considering the region of integration as the union of R −C, C−C∞ and ﬁnally C∞. The initial estimate was presented in [Wol90]. In Lemma A.4.1 of [Wol90, pg. 468] we showed −δ(z,z ) given δ0 > 0 there exists a constant c0 such that 0 1 provided the injectivity radius at z or z0 is at least δ0.

~~Lemma 12 For n>1, the operator (D − 2)−1 is a continuous mapping from C`n to C`.~~

Proof. We begin with the contribution to R Gg dA for the compact region R −C. Lemma A.4.1, [Wol90], provides that the integral is valued in C` −δ(z,z ) since for z0 ∈C, z ∈ R −C then e 0 ≤ `. The bound for the compact region is complete. The estimates for the cusp regions are indicated by considering the region C∞ at inﬁnity. We consider for z ∈C∞ the contribution

~~Z Z G(z, z0)g(z0) dA + G†(z, z0)g(z0) dA. C−C∞ C∞~~

For the first integral z ∈C∞ and z0 ∈C−C∞, while in the second integral z, z0 ∈C∞. On the indicated regions G and G† are smooth solutions of the differential equation (D − 2)u = 0. For z fixed and z0 tending to a puncture from Lemma A.4.1 each function tends to zero. It follows from the maximum principle that for z fixed G and G† achieve their maxima in z0 respectively on ∂(C−C∞) and ∂C∞. The estimate of Lemma A.4.1 applied for z0 respectively on ∂(C−C∞) and ∂C∞ now provides a uniform bound for G and G† by a constant multiple of the horocycle length function `. Since g is integrable on R the desired bounds for the integrals follow. It remains to consider the principal contribution G (z, z )g(z ) dA. RC∞ ∞ 0 0 For the upper half plane H the integral is unfolded and g is replaced by y−n

18 to give the integral Z −n−2 Q(z, z0)y dxdy (9) =z0≥1 for the variable z0 = x + iy. We introduce a majorant for Q. The values of −1 the function y on a disc {δ(z, z0) ≤ 1} are within ﬁxed multiples of the value at the center. It follows for sake of bounding (9) that for hyperbolic distance δ ≤ 1 the contribution of the kernel Q is bounded by a comparison kernel with a positive minimum on the disc. It now follows from the large −2δ(z,z ) hyperbolic distance description of the behavior −Q2(δ(z, z0)) ≤ ce 0 , 2 [Wol90, pg. 468], and the formula cosh δ(z, z )=1+|z−z0 | that the integral 0 2=z=z0 (9) is bounded for a = =z in terms of

~~− |z − ia|2 2 Z 1+ y−n−2dxdy −∞~~

a2 ≤ 1 −1 Upon substituting 3 a the last integral is bounded by O(a ). (y2+a2) 2 The integral (9) is bounded by a multiple of the horocycle length function ` = a−1, as desired. The proof is complete.

~~References~~

~~[AB60] Lars Ahlfors and Lipman Bers. Riemann’s mapping theorem for variable metrics. Ann. of Math. (2), 72:385–404, 1960.~~

~~[AC96] Enrico Arbarello and Maurizio Cornalba. Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves. J. Algebraic Geom., 5(4):705–749, 1996.~~

~~[Ahl61] Lars V. Ahlfors. Some remarks on Teichm¨uller’sspace of Riemann surfaces. Ann. of Math. (2), 74:171–191, 1961.~~

~~[Ahl73] Lars V. Ahlfors. Conformal invariants: topics in geometric function theory. McGraw-Hill Book Co., New York, 1973. McGraw- Hill Series in Higher Mathematics.~~

~~[Ber74] Lipman Bers. Spaces of degenerating Riemann surfaces. In Discontinuous groups and Riemann surfaces (Proc. Conf., Univ.~~

~~19 Maryland, College Park, Md., 1973), pages 43–55. Ann. of Math. Studies, No. 79. Princeton Univ. Press, Princeton, N.J., 1974.~~

~~[Bor97] Armand Borel. Automorphic forms on SL2(R), volume 130 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1997.~~

~~[DW03] Georgios Daskalopoulos and Richard Wentworth. Classiﬁcation of Weil-Petersson isometries. Amer. J. Math., 125(4):941–975, 2003.~~

~~[Ear77] Cliﬀord J. Earle. Teichm¨ullertheory. In Discrete groups and automorphic functions (Proc. Conf., Cambridge, 1975), pages 143– 162. Academic Press, London, 1977.~~

~~[Fay77] John D. Fay. Fourier coeﬃcients of the resolvent for a Fuchsian group. J. Reine Angew. Math., 293/294:143–203, 1977.~~

~~[Gar75] Frederick P. Gardiner. Schiﬀer’s interior variation and quasicon- formal mapping. Duke Math. J., 42:371–380, 1975.~~

~~[Gui94] Victor Guillemin. Moment maps and combinatorial invariants of Hamiltonian T n-spaces, volume 122 of Progress in Mathematics. Birkh¨auserBoston Inc., Boston, MA, 1994.~~

~~[KL05] Yon-Seo Kim and Kefeng Liu. A simple proof of Witten conjecture through localization. preprint, 2005.~~

~~[KL06] M. E. Kazarian and S. K. Lando. An algebro-geometric proof of Witten’s conjecture. preprint, 2006.~~

~~[Kon92] Maxim Kontsevich. Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys., 147(1):1–23, 1992.~~

~~[Leu67] Armin Leutbecher. Uber¨ Spitzen diskontinuierlicher Gruppen von lineargebrochenen Transformationen. Math. Z., 100:183–200, 1967.~~

~~[Mas76] Howard Masur. Extension of the Weil-Petersson metric to the boundary of Teichmuller space. Duke Math. J., 43(3):623–635, 1976.~~

20 [Mir07a] Maryam Mirzakhani. Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. Invent. Math., 167(1):179–222, 2007. [Mir07b] Maryam Mirzakhani. Weil-Petersson volumes and intersection theory on the moduli space of curves. J. Amer. Math. Soc., 20(1):1–23 (electronic), 2007. [MS06] Motohico Mulase and Brad Safnuk. Mirzakhani’s recursion relations, Virasoro constraints and the KdV hierarchy. preprint, 2006. [Mum77] David Mumford. Stability of projective varieties. Enseignement Math. (2), 23(1-2):39–110, 1977. [Nag88] Subhashis Nag. The complex analytic theory of Teichmüller spaces. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons Inc., New York, 1988. A Wiley-Interscience Publication. [Obi99] Kunio Obitsu. Non-completeness of Zograf-Takhtajan’s Kähler metric for Teichmüllerspace of punctured Riemann surfaces. Comm. Math. Phys., 205(2):405–420, 1999. [OP01] A. Okounkov and R. Pandharipande. Gromov-Witten theory, Hurwitz theory, and matrix models, I. preprint, 2001. [OTW06] Kunio Obitsu, Wing-Keung To, and Lin Weng. The asymptotic behavior of the Takhtajan-Zograf metrics. preprint, 2006. [OW07] Kunio Obitsu and Scott A. Wolpert. A degeneration expansion for the hyperbolic metric. preprint, 2007. [TZ87] L. A. Takhtadzhyan and P. G. Zograf. A local index theorem for families of ∂-operators on Riemann surfaces. Uspekhi Mat. Nauk, 42(6(258)):133–150, 248, 1987. [TZ91] L. A. Takhtajan and P. G. Zograf. A local index theorem for families of ∂-operators on punctured Riemann surfaces and a new Kählermetric on their moduli spaces. Comm. Math. Phys., 137(2):399–426, 1991. [Ven90] Alexei B. Venkov. Spectral theory of automorphic functions and its applications. Kluwer Academic Publishers Group, Dordrecht, 1990. Translated from the Russian by N. B. Lebedinskaya.

~~21 [Wen01] Lin Weng. Ω-admissible theory. II. Deligne pairings over moduli spaces of punctured Riemann surfaces. Math. Ann., 320(2):239– 283, 2001.~~

~~[Wit91] Edward Witten. Two-dimensional gravity and intersection theory on moduli space. In Surveys in diﬀerential geometry (Cambridge, MA, 1990), pages 243–310. Lehigh Univ., Bethlehem, PA, 1991.~~

~~[Wit92] Edward Witten. Two-dimensional gauge theories revisited. J. Geom. Phys., 9(4):303–368, 1992.~~

~~[Wol82] Scott A. Wolpert. The Fenchel-Nielsen deformation. Ann. of Math. (2), 115(3):501–528, 1982.~~

~~[Wol83a] Scott A. Wolpert. On the homology of the moduli space of stable curves. Ann. of Math. (2), 118(3):491–523, 1983.~~

~~[Wol83b] Scott A. Wolpert. On the K¨ahlerform of the moduli space of once punctured tori. Comment. Math. Helv., 58(2):246–256, 1983.~~

~~[Wol83c] Scott A. Wolpert. On the symplectic geometry of deformations of a hyperbolic surface. Ann. of Math. (2), 117(2):207–234, 1983.~~

~~[Wol85] Scott A. Wolpert. On the Weil-Petersson geometry of the moduli space of curves. Amer. J. Math., 107(4):969–997, 1985.~~

~~[Wol86] Scott A. Wolpert. Chern forms and the Riemann tensor for the moduli space of curves. Invent. Math., 85(1):119–145, 1986.~~

~~[Wol89] Michael Wolf. The Teichm¨ullertheory of harmonic maps. J. Diﬀerential Geom., 29(2):449–479, 1989.~~

~~[Wol90] Scott A. Wolpert. The hyperbolic metric and the geometry of the universal curve. J. Diﬀerential Geom., 31(2):417–472, 1990.~~

~~[Wol94] Scott A. Wolpert. Disappearance of cusp forms in special families. Ann. of Math. (2), 139(2):239–291, 1994.~~

[Wol03] Scott A. Wolpert. Geometry of the Weil-Petersson completion of Teichm¨ullerspace. In Surveys in Diﬀerential Geometry VIII: Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck, pages 357–393. Intl. Press, Cambridge, MA, 2003.

22 [Zog93] Peter Zograf. The Weil-Petersson volume of the moduli space of punctured spheres. In Mapping class groups and moduli spaces of Riemann surfaces (G¨ottingen,1991/Seattle, WA, 1991), volume 150 of Contemp. Math., pages 367–372. Amer. Math. Soc., Providence, RI, 1993.